In this paper, the existence and multiplicity of solutions of Kirchhoff type problems with critical nonlinearity is considered in \(\mathbb {R}^{3}: -\varepsilon ^{2}\left (a+b\displaystyle {\int }_{\mathbb {R}^{3}}|\nabla u|^{2}dx\right ){\Delta } u + V(x)u -\varepsilon ^{2}{\Delta }(u^{2})u = K(x)|u|^{22^{\ast }-2}u + h(x,u)\), \((t, x) \in \mathbb {R} \times \mathbb {R}^{3}\). Under suitable assumptions, we prove that it has at least one solution and for any \(m \in \mathbb {N}\), it has at least m pairs of solutions.

中文翻译：

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ℝ3$ \ mathbb {R} ^ {3} $中具有临界非线性的Kirchhoff型问题解的存在性

本文在\（\ mathbb {R} ^ {3}：-\ varepsilon ^ {2} \ left（a + b \ displaystyle {\ int } _ {\ mathbb {R} ^ {3}} | \ nabla u | ^ {2} dx \ right）{\ Delta} u + V（x）u-\ varepsilon ^ {2} {\ Delta}（u ^ {2}）u = K（x）| u | ^ {22 ^ {\ ast} -2} u + h（x，u）\），\（（t，x）\ in \ mathbb {R} \ times \ mathbb {R} ^ {3} \）。在适当的假设下，我们证明它具有至少一个解，对于任何\（m \ in \ mathbb {N} \），它至少具有m对解。